Sequence Of Successive Approximations Research Articles

Iterative Process for a Common Fixed Point of a Finite Family of Asymptotically k-Pseudocontractive Maps

Let $K$ be a closed convex nonempty subset of a Hilbert space $H$ and let $\{T_i\}_{i=1}^N$ be a finite family of Asymptotically k-pseudocontractive maps from $K$ into itself with $F = ∩_{i=1}^NF(T_i) not empty. Sufficient conditions for the strong convergence of the sequence of successive approximations generated by a Picard-like process to a common fixed point of the family are proved.

On Successive Approximations for Compact-Valued Nonexpansive Mappings

We show that for a given initial point the typical, in the sense of Baire category, nonexpansive compact valued mapping F has the following properties: there is a unique sequence of successive approximations and this sequence converges to a fixed point of F. In the case of separable Banach spaces we show that for the typical mapping there is a residual set of initial points that have a unique trajectory.

Strong Convergence Theorem for Finite Family of Asymptotically Nonexpansive in the Intermediate Sense Nonself Maps

Let K be a nonexpansive retract of a uniformly convex Banach space X with retraction P. Let Ti: K → X (i= 1,...,m) be a finite family of uniformly continuous asymptotically nonexpansive in the intermediate sense maps with a nonempty common fixed points set F. Sufficient conditions for the strong convergence of a sequence of successive approximations generated by an m-step algorithm to a point of F are proved.MSC(2010): 47H10, 47J25

Algorithm for Approximate Solving of a Nonlinear Boundary Value Problem for Generalized Proportional Caputo Fractional Differential Equations

In this paper an algorithm for approximate solving of a boundary value problem for a nonlinear differential equation with a special type of fractional derivative is suggested. This derivative is called a generalized proportional Caputo fractional derivative. The new algorithm is based on the application of the monotone-iterative technique combined with the method of lower and upper solutions. In connection with this, initially, the linear fractional differential equation with a boundary condition is studied, and its explicit solution is obtained. An appropriate integral fractional operator for the nonlinear problem is constructed and it is used to define the mild solutions, upper mild solutions and lower mild solutions of the given problem. Based on this integral operator we suggest a scheme for obtaining two monotone sequences of successive approximations. Both sequences consist of lower mild solutions and lower upper solutions of the studied problem, respectively. The monotonic uniform convergence of both sequences to mild solutions is proved. The algorithm is computerized and applied to a particular example to illustrate the theoretical investigations.

On generic convergence of successive approximations of mappings with convex and compact point images

We study the generic behavior of the method of successive approximations for set-valued mappings in separable Banach spaces. We consider the case of nonexpansive mappings with convex and compact point images and show that for the typical such mapping and typical points of its domain the sequence of successive approximations is unique and converges to a fixed point of the mapping.

A uniqueness criterion for ordinary differential equations

We show that a convex combination of the Nagumo and Osgood growth conditions for the spatial variation of continuous functions ensures the uniqueness of the solution to the initial-value problem for the associated ordinary differential equation. Moreover, this unique solution is obtained as the uniform limit of the sequence of successive approximations.

Approximation Approach to the Fractional BVP with the Dirichlet Type Boundary Conditions

We use a numerical-analytic technique to construct a sequence of successive approximations to the solution of a system of fractional differential equations, subject to Dirichlet boundary conditions. We prove the uniform convergence of the sequence of approximations to a limit function, which is the unique solution to the boundary value problem under consideration, and give necessary and sufficient conditions for the existence of solutions. The obtained theoretical results are confirmed by a model example.

?n the Analysis of Certain Flotation Processes with Velocities Depending on Time and Height of the Column

— The present paper is an extension of the previous paper of the author where the flotation column dynamics has been investigated. Here we consider the case when particle sedimentation rate and bubble lifting speed depend on time and position in the column. We use the methods for examining the transmission lines set out in the papers mentioned in the References. We formulate a mixed problem for the system describing the processes in the column and present it in a suitable operator form. Then we prove an existence - uniqueness of generalized solution by the fixed point method. We show an explicit approximated solution as a step in the sequence of successive approximations.

Оn the Analysis of Certain Flotation Processes with Velocities Depending on Time and Height of the Column

— The present paper is an extension of the previous paper of the author where the flotation column dynamics has been investigated. Here we consider the case when particle sedimentation rate and bubble lifting speed depend on time and position in the column. We use the methods for examining the transmission lines set out in the papers mentioned in the References. We formulate a mixed problem for the system describing the processes in the column and present it in a suitable operator form. Then we prove an existence - uniqueness of generalized solution by the fixed point method. We show an explicit approximated solution as a step in the sequence of successive approximations.

Successive approximations for a differential equation in a Banach space via Constantin condition

Abstract We give some sufficient conditions for the convergence of the sequence of successive approximations to the unique solution of first order Cauchy problem in a Banach space. Our approach is based on a generalized Nagumo condition due to A. Constantin and the properties of the Kuratowski measure of noncompactness.

On a class of stochastic integro-differential equations

ABSTRACT We prove the existence and uniqueness of a bounded random solution for a nonlinear stochastic integrodifferential equation on . We also show that the sequence of successive approximations converges to the solution almost surely and in mean square at each . An application of our result to a stochastic differential system is also given in Section 4.

ОN THE QUALITATIVE ANALYSIS OF CERTAIN FLOTATION PROCESSES

The present paper is devoted to the qualitative analysis of certain flotation processes describing by a first order hyperbolic system of partial differential equations. The system in question is like telegrapher equations. That is why, we use the methods for examining the transmission lines set out in the papers mentioned in the References. We formulate a mixed problem for this system with boundary conditions corresponding to the processes in the flotation cameras. We present the mixed problem for the hyperbolic system in a suitable operator form and prove an existence of generalized solution by fixed point method. One can obtain an explicit approximated solution as a step in the sequence of successive approximations.

Asymptotic regularity, fixed points and successive approximations

Let (M,d) be a metric space. In this paper we survey some of the most relevant results which relate the three concepts involved in the title: a) the asymptotic regularity; b) the existence (and uniqueness) of fixed points and c) the convergence of the sequence of successive approximations to the fixed point(s), for a given operator f : M ? M or for two operators f,g : M ? M connected to each other in some sense.

Strong Convergence Theorem for Finite Family of Generalised Asymptotically Nonexpansive Maps

Let K be a nonexpansive retract of a uniformly convex Banach space X with retraction P and Ti:1 · · ·,m: K-> X a finite family of uniformly continuous generalised asymptotically nonexpansive maps with a nonempty common fixed point set F. We provided and proved sufficient conditions for the strong convergence of a sequence of successive approximations generated by an m-step algorithm to a point of F.

The method of finite differences for nonlinear functional differential equations of the first order

Abstract Nonlinear functional partial differential equations with initial conditions are considered on the cone. The weak convergence of a sequence of successive approximations is proved. The proof is given by the duality principle.

Dhage iterative principle for quadratic perturbation of fractional boundary value problems with finite delay

In this article, by employing Dhage iterative method embodied in current hybrid fixed point theorem (HFPT) of Dhage, we derive an algorithm for the numerical solutions via construction of a sequence of successive approximations for a fractional order boundary value problem (FBVP) with finite delay. By using this technique, we obtain existence as well as approximation of solutions under weaker partial Lipschitz and partial compactness type conditions in a partially ordered Banach space. Additionally, we prove an existence and uniqueness theorem under a weaker partial nonlinear Lipschitz condition. The assumptions and main outcomes are also illustrated by two examples.

AN EFFICIENT STEP METHOD FOR A SYSTEM OF DIFFERENTIAL EQUATIONS WITH DELAY

Using the step method, we study a system of delay differential equations and we prove the existence and uniqueness of the solution and the convergence of the successive approximation sequence using the Perov's contraction principle and the step method. Also, we propose a new algorithm of successive approximation sequence generated by the step method and, as an example, we consider some second order delay differential equations with initial conditions.

Approximating Monotone Positive Solutions of a Nonlinear Fourth-Order Boundary Value Problem via Sum Operator Method

In this article, the authors investigate the existence and uniqueness as well as approximations of monotone positive solutions for a nonlinear fourth-order boundary value problem of the form \( u^{(4)}(t)=f(t,u(t)),\ 0< t<1; u(0)=u'(0)= u'(1)=0, u^{(3)}(1)+g(u(1))=0,\) where \(f\in C([0,1]\times [0,+\infty ),[0,+\infty )),\ g\in C([0,+\infty ),[0,+\infty )).\) It is shown that the above boundary value problem has a unique monotone positive solution and the sequence of successive approximations converges to the monotone positive solution under some proper conditions. These results are based upon two fixed point theorems of a sum operator in partial ordering Banach space. Finally, two examples are also given to illustrate the main abstract results.

Establishing motion control in children with autism and intellectual disability: Applications for anatomical and functional MRI.

Excessive motion makes magnetic resonance imaging (MRI) extremely challenging among children with autism spectrum disorder (ASD). The medical risks of sedation establish the need for behavioral interventions to promote motion control among children with ASD undergoing MRI scans. We present a series of experiments aimed at establishing both tolerance of the MRI environment and a level of motion control that would be compatible with a successful MRI. During Study 1, we evaluated the effects of prompting and contingent reinforcement on compliance with a sequence of successive approximations to an MRI using a mock MRI. During Study 2, we used prompting and progressive differential reinforcement of other behaviors (DRO) to promote motion control in a mock MRI for increasing periods of time. Finally, during Study 3, some of the participants underwent a real MRI scan while a detailed in-session motion analysis informed the quality of the images captured.

Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes

In this paper the existence and uniqueness of the solution for a stochastic nonlinear Schrödinger equation, which is perturbed by a cylindrical Wiener process is investigated. The existence of the variational solution and of the generalized weak solution are proved by using sequences of successive approximations, which are the solutions of certain linear problems.